Monitoring Synchronization of Regional Recessions:
A Markov-Switching Network Approach�
[JOB MARKET PAPER]
Danilo Leiva-Leony
University of Alicante
Abstract
This paper develops a new framework for monitoring changes in the degree of
synchronization between many stochastic processes subject to regime changes. In
the proposed methodology the estimated time-varying dependence relations among
the hidden Markov processes governing the system can be interpreted as a dynamic
weighted network. Bayesian estimates of an empirical application to the synchroniza-
tion of business cycle phases in U.S. states suggest that national recessions can be an-
ticipated by an index that accounts for the global synchronization between states, con-
�rming its predictive ability with real-time exercises. Moreover, the way in which an
upcoming national recession could simultaneously a¤ect each of its smaller economies
at the state level can be dynamically evaluated.
Keywords: Regional Business Cycle, Markov-Switching, Network Analysis.
JEL Classi�cation: E32, R12, C32, C45.
�I thank Gabriel Perez-Quiros and Maximo Camacho for valuable comments and suggestions which
have greatly improved the content of this paper. Supplementary material of this paper can be found at
the author�s webpage: https://sites.google.com/site/daniloleivaleon/mediayCorresponding author. Departamento de Fundamentos del Análisis Económico, Universidad de Ali-
cante, Campus San Vicente del Raspeig 03690, Alicante - Spain. Phone: (+34) 965903400 (ext. 3026).
1
1 Introduction
The interest in identifying changes in business cycles synchronization started to markedly
increase since the implementation of the European Monetary Union, due to more synchro-
nized countries are expected to face smaller costs of joining the Union than those countries
with relatively less synchronized cycles, Camacho et al. (2006). In other words, analyzing
synchronization changes is crucial for policy makers in order to determine which countries,
regions or even sectors of an economy could be more sensitive to global policy shocks and
which others could remain less a¤ected by them.
Since the seminal work by Hamilton (1989) in which U.S. business cycle phases are
characterized by using a Markov-switching (MS) model, a broad range of extensions related
to this approach have been developed due to its great success. In particular, multivariate
MS models have become a useful tool in analyzing synchronization between the business
cycle phases of di¤erent countries (Smith and Summers, 2005 and Camacho and Perez-
Quiros, 2006) or regions (Owyang et al., 2005 and Hamilton and Owyang, 2010). Although
all these studies provide a picture about how much some business cycles are in sync during
a given period, they are not able to capture changes in the degree of synchronization that
can occur due to economic unions, policy changes, or even aggregate contractionary shocks.
This is because in order to preserve parsimony in the model, a still non explored question
that could help to unveil this feature has remained unnoticed: What is the dynamic
relationship between the unobserved state variables governing a multivariate MS setting?
The approaches followed in the literature traditionally assume a �xed over time depen-
dence relation between such state variables that can be divided into two categories. There
are studies in which such relation is just a priori assumed based on the econometrician�s
judgement. Apart from the general Markovian speci�cation, which involves the estimation
of the full transition probability matrix,1 multivariate MS models are usually analyzed un-
der three di¤erent types of relationships between the unobserved state variables governing
1This approach presents computational di¢ culties as the model increases in the number of series, states
or lags.
2
each time series, see Hamilton and Lin (1996) and Anas et al. (2007). The �rst one refers
to the case in which all series follow a common regime dynamics, Krolzig (1997). Second,
the use of totally independent Markov chains, which is the most followed approach, Smith
and Summers (2005) and Chauvet and Senyuz (2008). Third, the dynamics in one state
variable precede those of other state variables, Hamilton and Perez-Quiros (1996) and
Cakmakli et al. (2011).
Other studies focus, on the one hand, in obtaining a degree of synchronization be-
tween MS processes for a given sample period, providing average dependence relationship
estimates, as in the case of Artis et al. (2004) who compute cross-correlations between
the smoothed state probabilities after estimating several univariate models. On the other
hand, some works focus on testing synchronization by relying on the extreme cases of in-
dependence and perfect synchronization. Some examples are Harding and Pagan (2006),
who propose tests of the hypotheses that cycles are either unsynchronized or perfectly syn-
chronized under complications caused by serial correlation and heteroscedasticity in cycle
states, and Pesaran and Timmermann (2009) who test independence between discrete
multicategory variables based on canonical correlations. One intuitive approach followed
by Camacho and Perez-Quiros (2006), Bengoechea et al. (2006), and Leiva-Leon (2011),
is based on modeling the data generating process as a linear combination between the two
polar cases, claiming that in most of real situations, the true MS multivariate dynamics
should be somewhere in between them. Although all these previous approaches can be
used to study how much synchronized are a set of MS processes for a given period of time,
they are not able to identify possible changes in synchronization of the state variables.
This paper proposes a novel framework for monitoring changes in the degree of ag-
gregate synchronization between many stochastic processes that are subject to regime
changes. Speci�cally, it computes regime inferences from a multivariate Markov-switching
model and simultaneously it obtains a measure of the time-varying synchronization be-
tween the unobserved state variables governing each process. Such measure is endoge-
nously estimated as a weighted average between the dependent and independent polar
cases by making inference on the regimes of high and low synchronization without requir-
ing a posteriori computations as in the case of the cooncordance in Harding and Pagan
3
(2006). One advantage of this approach in comparison to the conventional dynamic corre-
lation measure, is that pairwise synchronizations, which are estimated through Bayesian
methods, can be easily converted into desynchronization measures to be combined with
dynamic multidimensional scaling, and network analysis in order to provide assessments
regarding to possible changes in the clustering patterns that could experiment a system
of time series, the key components leading the system, and even to make inference about
its future behaviour.
The proposed framework is used as a tool for monitoring the time-varying synchroniza-
tion among U.S. states business cycle phases in order to assess, �rst, up to which extend
the interconnectedness between smaller economies can be helpful to anticipate national
recessions? and second, how an upcoming national recession could simultaneously a¤ect
each of its smaller economies at the state level?
In a network where U.S. states are interpreted as nodes and the strength of the links
between nodes is given by the degree of business cycle synchronization between two states,
an index of global prominence or centrality of U.S. is computed. It shows a markedly
high tendency to increase some months before national recessions take place, keeping
high values during the whole contractionary episode and even some months after it ends,
then returning to a stable level that seems to prevail during the rest of the expansion
period. This agrees with the premise that when U.S. states business cycles start to closely
move in the same way, the economy becomes more fragile. The reliability of this index
to anticipate recessions is con�rmed with real time exercises by reestimating it with the
available information up to some period before national recessions take place.
The interdependence among states and national business cycles is assessed relying on
a network analysis. The clustering coe¢ cient and closeness centrality measures are used
to make time-varying assessments of the comovement among economic phases, reporting
high values when economies are following the same pattern (recessions or expansions),
and dropping when their phases are independent of each other by following idiosyncratic
behaviors. Since the degree of interdependence between states can be monitored month-
to-month, it is possible to evaluate how much state "A" could be a¤ected by economic
shocks hitting state "B" or the nation as a whole.
4
Additionally, the study provides three noteworthy features that deserve attention.
First, it reports abrupt changes in business cycle sync phases between U.S. states, specif-
ically, before the mid 90�s where the cohesiveness between states was cyclically changing
between high and low levels, followed by a period of relative stability at high levels that
has prevailed after that date.
Second, the set of economies with the highest centrality is composed by the most
central state, North Carolina, followed by Missouri, Wisconsin, Tennessee, and Alabama.
Interestingly, these �ve states are also geographically linked, constituting a region of the
U.S. business cycle core in terms of synchronization. This core is located in the inter-
section among three of the eight BEA regions, Southeast, Great Lakes and Plains, and
can be interpreted as the leading dynamics of the national economy, which is useful for
anticipating contractionary episodes.
Third, apart from dynamic estimates of the interdependence degree, this framework
also provides the possibility of computing stationary estimates. They are obtained with
the ergodic or time-independent probabilities associated to the latent variable measuring
synchronization. This stationary results report that states in the core roughly coincide
with the ones showing high cooncordance with the national business cycle found in Owyang
et al. (2005), that were obtained under a univariate approach. Moreover, U.S. states can
be grouped into three clusters, a highly, discreetly and lowly in sync with the national
business cycle, result that agrees with the one in Hamilton and Owyang (2010).
The paper is structured as follows. Section 2 provides the Markov-switching synchro-
nization modelling approach, the �ltering algorithm and the procedure of the Bayesian
parameter estimation. Section 3 analyzes the business cycle phases synchronization in
U.S. states relying on network analysis, providing multidimensional scaling, clustering
coe¢ cient, and closeness centrality estimates. Section 4 concludes.
2 Modelling Markov-Switching Synchronization
In this section, it is proposed an algorithm to monitor changes in the degree of pairwise
synchronization between stochastic processes that are subject to regime changes. Let yi;t
5
be a time series modeled as a function of a latent variable, Si;t, that indicates the regime
at which it is, an idiosyncratic component �i;t, and a set of parameters, �i, to be estimated.
Accordingly, for i = a; b,
ya;t = f(Sa;t; �a;t; �a) (1)
yb;t = f(Sb;t; �b;t; �b); (2)
the goal will be to make an assessment on their synchronization for each period of time,
that is,
�abt = sync(Sa;t; Sb;t) = Pr(Sa;t = Sb;t): (3)
Speci�cally, this paper will focus on the time-varying sync between the two unobserved
state variables governing a bivariate Markov-switching model. In order to mainly focus on
modelling this dynamic dependence relation and to avoid complex notation, it is consid-
ered the following parsimonious and very tractable bivariate two-state Markov-switching
speci�cation:24 ya;t
yb;t
35 =24 �a;0 + �a;1Sa;t
�b;0 + �b;1Sb;t
35+24 "a;t
"b;t
35 ;24 "a;t
"b;t
35 � N
0@24 00
35 ;24 �2a �ab
�ab �2b
351A ;
(4)
The results obtained in this section can be straightforwardly applied to an extended speci-
�cation that could includes more lags in the dynamics or even Markov-switching variance-
covariance matrix. The state variable Sk;t indicates if ykt is in regime 0 with a mean
equal to �k;0 , where Sk;t = 0, or if ykt is in regime 1 with a mean equal to �k;1, where
Sk;t = 1, for k = a; b. Moreover Sa;t and Sb;t evolve according to irreducible two-state
Markov chains, whose transition probabilities are given by
Pr(Sk;t = jjSk;t�1 = i) = pk;ij , for i; j = 0; 1 and k = a; b: (5)
In order to characterize the dynamics of yt = [ya;t; yb;t]0, the information contained in
Sa;t and Sb;t can be summarized in the state variable, Sab;t, it will account for the possible
combinations that the vector, �Sab;t =��a;0 + �a;1Sa;t; �b;0 + �b;1Sb;t
�0, could take trough
6
the di¤erent regimes. It is de�ned as:
Sab;t =
8>>>>>><>>>>>>:
1, If Sa;t = 0; Sb;t = 0
2, If Sa;t = 0; Sb;t = 1
3, If Sa;t = 1; Sb;t = 0
4, If Sa;t = 1; Sb;t = 1
: (6)
In contrast to the previous ways of modelling Pr(Sab;t = jab) in the literature, where
some exogenous prior speci�c relationship between Sa;t and Sb;t is assumed, this paper
proposes to model Pr(Sab;t = jab) based on the individual dynamics of Sa;t and Sb;t,
and simultaneously accounting for the dynamic degree of dependence between each other,
which is endogenously estimated.
Although the degree of dependence between Sa;t and Sb;t is unknown, the two opposite
extreme cases of dependence relationships are known. That is, on the one hand, if Sa;t
and Sb;t are fully independent, then Pr(Sa;t = ja; Sb;t = jb) = Pr(Sa;t = ja) Pr(Sb;t = jb).
On the other hand, if Sa;t and Sb;t are totally dependent, in the sense that they are fully
synchronized, then ya;t and yb;t are driven by the same state variable, St, i.e. Sa;t = Sb;t =
St, remaining in this case Pr(Sa;t = ja; Sb;t = jb) = Pr(St = j). In empirical applications,
the true dependence degree should be located between these two extreme possibilities.
In order to make inference on the type of dependence between Sa;t and Sb;t, a new
unobserved state variable will be de�ned as:
Vt =
8<: 0 If Sa;t and Sb;t are fully independent
1 If Sa;t and Sb;t are totally dependent; (7)
In order to maintain the nonlinear nature of the framework, it will also evolve according
to an irreducible two-state Markov chain whose transition probabilities are given by
Pr(Vt = jvjVt�1 = iv) = pv;kl, for iv; jv = 0; 1 (8)
Model in Equation (4) hence remains fully characterized by the state variable S�t ,
which collects information regarding to joint dynamics, individual dynamics and their
7
dependence relationship over time simultaneously. In this way S�ab;t is de�ned as:2
S�ab;t =
8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:
1, If Vt = 0; Sa;t = 0; Sb;t = 0
2, If Vt = 0; Sa;t = 0; Sb;t = 1
3, If Vt = 0; Sa;t = 1; Sb;t = 0
4, If Vt = 0; Sa;t = 1; Sb;t = 1
5, If Vt = 1; Sa;t = 0; Sb;t = 0
6, If Vt = 1; Sa;t = 0; Sb;t = 1
7, If Vt = 1; Sa;t = 1; Sb;t = 0
8, If Vt = 1; Sa;t = 1; Sb;t = 1
; (9)
Inference on the possible states of S�ab;t, i.e. Pr(S�ab;t = j�ab) for j
�ab = 1; : : : ; 8, can be
done by computing
Pr(Sa;t = ja; Sb;t = jb; Vt = jv) = Pr(Sa;t = ja; Sb;t = jbjVt = jv) Pr(Vt = jv) (10)
where Pr(Sa;t = ja; Sb;t = jbjVt = jv) indicates the inference on the dynamics of Sab;t
conditional on total independence, Vt = 0, or conditional on full dependence, Vt = 1. In
the former case the joint probability of S�ab;t will be
Pr(Sa;t = ja; Sb;t = jb; Vt = 0) = Pr(Sa;t = ja; Sb;t = jbjVt = 0)Pr(Vt = 0)
= Pr(Sa;t = ja) Pr(Sb;t = jb) Pr(Vt = 0); (11)
and in the latter case, it will be
Pr(Sa;t = ja; Sb;t = jb; Vt = 1) = Pr(Sa;t = ja; Sb;t = jbjVt = 1)Pr(Vt = 1)
= Pr(St = j) Pr(Vt = 1); (12)
therefore probabilities of the state variable Sab;t in Equation (6) after accounting for syn-
chronization, can be easily computed as
Pr(Sa;t = ja; Sb;t = jb) = Pr(Vt = 1)Pr(St = j)+(1�Pr(Vt = 1))Pr(Sa;t = ja) Pr(Sb;t = jb);
(13)
2States 6 and 7 in Equation (9) are truncated to zero by construction, since the two state variables can
not be in di¤erent states if they are perfectly synchronized, i.e. Pr(Sa;t = ja; Sb;t = jbjVt = 1) = 0 for any
ja 6= jb.
8
which indicates that joint dynamics of Sa;t and Sb;t are characterized by a linear combi-
nation between the extreme dependent case and the extreme independent case, where the
weights assigned to each of them are endogenously determined by their sync degree
�abt = Pr(Vt = 1): (14)
2.1 Filtering Algorithm
Following the line of Hamilton�s (1994) algorithm, I propose an extension to estimate the
model described in Equations (4) and (13). The algorithm is composed by two uni�ed
steps, in the �rst one the goal is the computation of the likelihoods, while in the second
one the goal is to compute prediction and updating probabilities.
STEP 1: For the moment it is assumed that model�s parameters are known and col-
lected in the vector � = (�a;0; �a;1; �b;0; �b;1; �2a; �
2b ; �ab; pa;00; pa;11; pb;00; pb;11; p00; p11; pv;00; pv;11)
0,
in the next section the Bayesian procedure to estimate � will be clari�ed. By using
the prediction probabilities3 Pr(Sk;t = jkj t�1; �) for k = a; b, Pr(Vt = jvj t�1; �) and
Pr(St = jj t�1; �), the joint probability corresponding to the state variable that fully
characterizes the model dynamics, S�ab;t, can be obtained relying on Equations (11) and
(12), that is
Pr(S�ab;t = j�abj t�1; �) = Pr(Sa;t = ja; Sb;t = jb; Vt = jvj t�1; �)
= Pr(Sa;t = ja; Sb;t = jbjVt = jv; t�1; �) Pr(Vt = jvj t�1; �); (15)
thereafter the density of yt given that it is on regime S�t , and the prediction probabilities
of the realizations of S�t , are used to compute the joint likelihood
f(yt; S�ab;t = j�abj t�1; �) = f(ytjS�ab;t = j�ab; t�1; �) Pr(S
�ab;t = j�abj t�1; �)
= f(yt; Sa;t = ja; Sb;t = jb; Vt = jvj t�1; �); (16)
then summing across the respective terms, speci�c likelihoods corresponding individual
3The steady state or ergodic probabilities can be used as starting values of the �lter.
9
processes are computed
fa(ya;t; Sa;t = jaj t�1; �) =1X
jb=0
1Xjv=0
f(yt; Sa;t = ja; Sb;t = jb; Vt = jvj t�1; �) (17)
fb(yb;t; Sb;t = jbj t�1; �) =1X
ja=0
1Xjv=0
f(yt; Sa;t = ja; Sb;t = jb; Vt = jvj t�1; �) (18)
fab(yt; St = jj t�1; �) =Xja=jb
f(yt; Sa;t = ja; Sb;t = jb; Vt = jvj t�1; �) (19)
f(yt; Vt = jvj t�1; �) =1X
ja=0
1Xjb=0
f(yt; Sa;t = ja; Sb;t = jb; Vt = jvj t�1; �): (20)
STEP 2: Once yt is observed at the end of time t, the prediction probabilities Pr(Sk;t =
jkj t�1; �) for k = a; b, Pr(Vt = jvj t�1; �) and Pr(St = jj t�1; �) can be updated as
follows
Pr(Sa;t = jaj t; �) =fa(ya;t; Sa;t = jaj t�1; �)
f(ytj t�1; �)(21)
Pr(Sb;t = jbj t; �) =fb(yb;t; Sb;t = jbj t�1; �)
f(ytj t�1; �)(22)
Pr(St = jj t; �) =f(yt; St = jj t�1; �)
f(ytj t�1; �)(23)
Pr(Vt = lj t; �) =f(yt; Vt = lj t�1; �)
f(ytj t�1; �)(24)
Where t = f t�1; ytg, and the unconditional likelihood function is given by
f(ytj t�1; �) =X8
j�=1f(yt; S
�ab;t = j�abj t�1; �): (25)
Forecasts of the updated probabilities in Equations (21)-(24) are done by using the cor-
responding transition probabilities in the vector �, that is pa;ij ; pb;ij ; pij ; pv;ij for Sa;t; Sb;t;
St; Vt respectively,
Pr(Sk;t+1 = jkj t; �) =1X
ik=0
Pr(Sk;t+1 = jk; Sk;t = ikj t; �)
=1X
ik=0
Pr(Sk;t+1 = jkjSk;t = ik) Pr(Sk;t = ikj t; �), for k = a; b (26)
10
Pr(Vt+1 = jvj t; �) =1Xi=0
Pr(Vt+1 = jv; Vt = ivj t; �)
=1Xi=0
Pr(Vt+1 = jvjVt = iv) Pr(Vt = ivj t; �) (27)
Pr(St+1 = jj t; �) =1Xi=0
Pr(St+1 = j; St = ij t; �)
=
1Xi=0
Pr(St+1 = jjSt = i) Pr(St = ij t; �). (28)
Finally the above forecasted probabilities are used to predict inferences on the realizations
of S�ab;t+1, again relying on Equations (11) and (12)
Pr(S�ab;t+1 = j�abj t; �) = Pr(Sa;t+1 = ja; Sb;t+1 = jb; Vt+1 = jvj t; �)
= Pr(Sa;t = ja; Sb;t = jbjVt = jv; t; �) Pr(Vt = jvj t; �); (29)
By iterating these two steps for t = 1; 2; : : : ; T , the algorithm simultaneously provides
inferences on the joint dynamics and individual dynamics along with their time-varying
degree of dependence of the model in Equations (4) and (13).
2.2 Bayesian Parameter Estimation
The approach to estimate � will be relied on a bivariate extended version of the multi-move
Gibbs-sampling procedure implemented by Kim and Nelson (1998) for Bayesian estimation
of univariate Markov-switching models.4 In this setting both the parameters of the model
� and the Markov-switching variables ~Sk;T = fSk;tgT1 for k = a; b, ~ST = fStgT1 and
~VT = fVtgT1 are treated as random variables given the data in ~yT = fytgT1 . The purpose of
this Markov chain Monte Carlo simulation method is to approximate the joint and marginal
distributions of these random variables by sampling from conditional distributions.
4The motivation for the use of Bayesian methods relies on the fact that as the number of possible states
increase, the likelihood function could be characterized by many local maxima and there could be strong
convergence problems in performing maximum likelihood estimation, Boldin (1996).
11
2.2.1 Priors
For the mean and variance parameters in vector �, the Independent Normal-Wishart prior
distribution is used
p(�;��1) = p(�)p(��1); (30)
where
� � N(�; V �)
��1 � W (S�1; �)
for the transition probabilities pa;00; pa;11 from Sa;t, pb;00; pb;11 from Sb;t, p00; p11 from St
and pv;00; pv;11 from Vt, Beta distributions will be used as conjugate priors
pk;00 � Be(uk;11; uk;10), pk;11 � Be(uk;00; uk;01), for k = a; b (31)
p00 � Be(u11; u10), p11 � Be(u00; u01) (32)
pv;00 � Be(uv;11; uv;10), pv;11 � Be(uv;00; uv;01) (33)
2.2.2 Drawing ~Sa;T , ~Sb;T , ~ST and ~VT given � and ~yT
Following the result in Equation (13), in order to make inference on the bivariate dynamics
of the model (4) driven by ~Sab;T = fSab;tgT1 and described in (6), it is just needed to make
inference on the dynamics of the single state variables ~Sa;T , ~Sb;T , ~ST and ~VT , this can be
done following the results in Kim and Nelson (1998) by �rst computing draws from the
conditional distributions
g( ~Sk;T j�; ~yT ) = g(Sk;T j~yT )TYt=1
g(Sk;tjSk;t+1; ~yt), for k = a; b (34)
g( ~ST j�; ~yT ) = g(ST j~yT )TYt=1
g(StjSt+1; ~yt) (35)
g( ~VT j�; ~yT ) = g(VT j~yT )TYt=1
g(VtjVt+1; ~yt): (36)
In order to obtain the two terms in the right hand side of Equation (34)-(35) the following
two steps can be employed:
12
Step 1: The �rst term can be obtained by running the �ltering algorithm developed
in Section 2.1, to compute g( ~Sk;tj~yt) for k = a; b, g( ~Stj~yt) and g( ~Vk;tj~yt) for t = 1; 2; : : : ; T ,
saving them and taking the elements for which t = T .
Step 2: The product in the second term can be obtained for t = T � 1; T � 2; : : : ; 1,
by following the result:
g(Stj~yt; St+1) =g(St; St+1j~yt)g(St+1j~yt)
/ g(St+1jSt)g(Stj~yt), (37)
where g(St+1jSt) corresponds to the transition probabilities of St and g(Stj~yt) were
saved in Step 1.
Then, it is possible to compute
Pr[St = 1jSt+1; ~yt] =g(St+1jSt = 1)g(St = 1j~yt)P1j=0 g(St+1jSt = j)g(St = jj~yt)
; (38)
and generate a random number from a U [0; 1]. If that number is less than or equal to
Pr[St = 1jSt+1; ~yt], then St = 1, otherwise St = 0. The same procedure applies for Sa;t,
Sb;t and Vt, and by using Equation (13) inference of ~Sab;T can be done.
2.2.3 Drawing pa;00,pa;11,pb;00,pb;11, p00,p11,pv;00,pv;11 given ~Sa;T , ~Sa;T , ~ST and ~VT
Conditional on ~Sk;T for k = a; b, ~ST and ~VT , the transition probabilities are independent
on the data set and the model�s parameters, hence the likelihood function of p00, p11is
given by:
L(p00; p11j ~ST ) = pn0000 (1� pn0100 )p
n1111 (1� p
n1011 ); (39)
where nij refers to the transitions from state i to j, accounted for in ~ST .
Combining the prior distribution in Equation (32) with the likelihood, the posterior
distribution is given by
p(p00; p11j ~ST ) / pu00+n00�100 (1� p00)u01+n01�1pu11+n11�111 (1� p11)u10+n10�1 (40)
which indicates that draws of the transition probabilities will be taken from
p00j ~ST � Be(u00 + n00; u01 + n01); p11j ~ST � Be(u11 + n11; u10 + n10) (41)
13
2.2.4 Drawing �0;a,�1;a,�0;b,�1;b given �2a,�2b ,�ab, ~Sa;T , ~Sb;T , ~ST , ~VT and ~yT
The model in Equation (4) can be compactly expressed as
24 ya;t
yb;t
35 =
24 10
Sa;t
0
0
1
0
Sb;t
3526666664�a;0
�a;1
�b;0
�b;1
37777775+24 "a;t
"b;t
35 ;24 "a;t
"b;t
35 � N
0@24 00
35 ;24 �2a
�ab
�ab
�2b
351A
yt = �St�+ �t; �t � N(0;�); (42)
stacking as:
y =
26666664y1
y2...
yT
37777775 ; �S =26666664�S1
�S2...
�ST
37777775 ; and � =26666664�1
�2...
�T
37777775 ;
the model in Equation (42) remains written as a normal linear regression model with an
error covariance matrix of a particular form:
y = S�+ �; � � N(0; I �) (43)
Conditional on the covariance matrix parameters, state variables and the data, by
using the corresponding likelihood function, the conditional posterior distribution
p(�j ~Sa;T ; ~Sb;T ; ~ST ; ~VT ;��1; ~yT ) takes the form
�j ~Sa;T ; ~Sb;T ; ~ST ; ~VT ;��1; ~yT � N(�; V �); (44)
where
V � =
V �1� +
TXt=1
�S0t��1 �St
!�1
� = V �
V �1� �+
TXt=1
�S0t��1yt
!:
After drawing � =h�a;0 �a;1 �b;0 �b;1
i0from the above multivariate distribution,
if the generated value of �a;1 or �b;1 is less than or equal to 0, that draw is discarded,
otherwise it is saved, this is in order to ensure that �a;1 > 0 and �b;1 > 0.
14
2.2.5 Drawing �2a,�2b ,�ab given �0;a,�1;a,�0;b,�1;b, ~Sa;T , ~Sb;T , ~ST , ~VT and ~yT
Conditional on the mean parameters, state variables and the data, by using the correspond-
ing likelihood function, the conditional posterior distribution p(��1j ~Sa;T ; ~Sb;T ; ~ST ; ~VT ; �; ~yT )
takes the form
��1j ~Sa;T ; ~Sb;T ; ~ST ; ~VT ; �; ~yT �W (S�1; �); (45)
where
� = T + �
S = S +TXt=1
�yt � �St�
� �yt � �St�
�0;
after ��1 is generated the elements is � are recovered.
3 Business Cycle Phases Synchronization in U.S. States
The global �nancial crisis has stimulated the interest on the sources and propagation
of contractionary episodes, calling to take a more careful look at the disaggregation of
the business cycle in order to study the mechanisms underlying economic �uctuations.
Two recent works have shown interesting features about the business cycle phases in U.S.
states, Owyang et al. (2005), and the propagation of regional recessions in the same
country, Hamilton and Owyang (2010). On the one hand, the former study that follows
a univariate approach, �nds that U.S. states di¤er signi�cantly in the timing of switches
between regimes of expansions and recessions, indicating large di¤erences in the extent
to which state business cycle phases are in concord with those of the aggregate economy.
On the other hand, the later work which follows a multivariate approach, focuses on
clustering the states sharing similar business cycle characteristics �nding that di¤erences
across states appear to be a matter of timing and they can be grouped into three clusters
with some of them entering in recession or recovering before others.
Although these previous studies provide insights about the synchronization among
di¤erent business cycles during a given period, they are not able to capture changes in
the degree of synchronization between the phases of economic �uctuations, that can be
15
potentially caused by economic unions, policy changes, aggregate recessionary shocks, etc.
Analyzing synchronization changes is crucial for policy makers in order to determine at
every period of time, which states could be more sensitive to speci�c policy or recessionary
shocks and which others would remain less a¤ected.
The framework developed in Section 2 is applied for monitoring monthly changes in
the degree of sync among economic phases of U.S. states, and also between states and
the national business cycle. The state coincident indexes constructed in Crone (2002) and
provided by the Federal Reserve Bank of Philadelphia, are used as indicators of economic
activity at state level. Alaska and Hawaii are excluded, as in Hamilton and Owyang (2010),
and The Chicago Fed National Activity Index (CFNAI) is used as monthly measure of
the aggregate U.S. business cycle, the sample period is 1979:8 to 2012:3.
3.1 Bivariate Synchronization
The analysis that was performed for 48 U.S. states required to model each of the C482 =
1; 128 pairwise comparisons, where in each of them 12,000 draws were generated from the
posterior with an initial burn of 2,000 draws that were discarded.5 In order to assess
the performance of the proposed MS synchronization model, two selected examples are
analyzed.6
The �rst one focuses on the case of two states which presents high and similar percent-
age of national GDP, that is, New York with 7.68% and Texas with 7.95%. Table 1 shows
the posterior means and medians for the model parameters along with their corresponding
standard deviation. All means and medians of parameters are similar and also statistically
signi�cant. It is worth to highlight the estimates of the transition probabilities associated
to the state variable that measures synchronization, Vt. For the New York vs. Texas
case, the probability of going from a regime of high sync to another regime of high sync
is almost equal to the probability of going from a regime of low sync to another regime
of low sync, since both are about 0.95. These estimates are corroborated in Chart A of
5Due to the high technology nowadays available, such high number of computations is not a problem
anymore.6The results for the other cases are available upon request to the author.
16
Figure 1, where the probabilities of recession for New York and Texas along with their
synchronization are plotted. As can be seen during the �rst half of the sample, both states
were lowly synchronized, however since 1994 they started to experiment high sync levels.
The second example focuses on analyzing two states of di¤erent GDP share by selecting
the polar case, that is, the state with the highest share, California with 13.34%, along
with the state with the lowest share, Vermont with 0.18%. Table 2 presents the Bayesian
estimation results of the model parameters. Unlike to the previous case, in the California
vs. Vermont model, the probability of remaining in a high sync regime (0.97) is higher
than the probability of remaining in a low sync regime (0.93). This can be observed in
Chart B of Figure 1, since with the exception of the 1987-1994 period, both states have
experimented high levels of synchronization among their business cycle phases.
These two examples, apart from documenting the existence of abrupt changes in busi-
ness cycle sync phases between U.S. states, show that their synchronization is independent
on their GDP share, since the proposed approach only focuses on making a time-varying
assessment of the comovement among economic phases, reporting high values when both
are following the same pattern (recessions or expansions), and dropping when their phases
are independent of each other by following idiosyncratic behaviors.
The analysis regarding to the cooncordance between states and national recessions is
performed in Owyang et al. (2005) by following the line in Harding and Pagan (2006),
where the degree to which two business cycles are in sync is calculated as the percentage
of time the two economies were in the same regime. However, such approach provides
an average synchronization measure without the possibility of making inference on the
potential changes that it could experiment trough time. The approach in this paper is
used to analyze this issue. Figure 2 shows the dynamic sync degree between states and
national business cycle, showing a high heterogeneity across states. On the one hand,
some of them show an almost constant and low sync degree, e.g. Louisiana, Oklahoma
and Wyoming, while others an almost constant and high sync degree, e.g. Alabama,
Minnesota, Nebraska, South Carolina, Tennessee and Vermont. On the other hand, the
rest of states in general experiment abrupt sync changes, that usually take place before or
after national recessions, as in the case of Massachusetts and North Dakota respectively
17
for the 1990�s recession, and North Carolina for 2007�s recession, these features agree with
Hamilton and Owyang (2010) who states that di¤erences across states appear to be a
matter of timing.
By means of the performed analysis, it can be evaluated how much state "A" could
be a¤ected by an economic shock that is hitting state "B", or which states could be more
sensitive when a period of national recession is coming, since the degree of interdependence
between states and national economic activity can be monitored month-to-month. How-
ever, an aggregate dynamic representation of the results presented so far seems needed in
order to deal with the question: How an upcoming national recession could simultaneously
a¤ect each of its smaller economies at the state level?
As suggested by Timm (2002) and Camacho et al. (2006), multidimensional scaling
methods are a helpful tool to identify cyclical a¢ liations between economies. Although,
traditionally studies focus in providing a map of just one general picture of the business
cycle similarities for a given sample period, for the present case a dynamic approach is
needed.
3.2 Dynamic Multidimensional Scaling
This methodology seeks to �nd a low dimensional coordinate system to represent n-
dimensional objects and create a map of lower dimension (k) which gives approximate
distances among them. The dimensional coordinates of the projection of any two objects,
a and b, are computed by minimizing a stress function which measure the squared sum of
divergences between the true distances or the measure of desynchronization, abt = 1��abt ,
and the approximate distances, ~ abt , among these objects. Moreover given that this repre-
sentation can be obtained for each t = 1; : : : ; T , it is applied a dynamic version, in which
the evolution of states synchronization is presented by a discrete-time sequence of graph
snapshots. In order to preserve the �mental map�between snapshots so that the transition
between frames in the animation can be easily interpreted, the movements between time
steps are constrained by adding a temporal penalty to the stress function of a static graph
18
layout method, Xu et al. (2011), so that the minimization problem is:
min~ abt=
nPa=1
nPb=1
( abt � ~ abt )2Pa;b(
abt )
2+ �
nXa=1
~ atjt�1; (46)
where � can be chosen as the average standard deviations of the series in order to adjust
the movements according to the variability of the data and with
~ abt = (jjza;t � zb;tjj2)1=2 ="kXi=1
(zai;t � zbi;t)2#1=2
(47)
~ atjt�1 = (jjza;t � za;t�1jj2)1=2 ="kXi=1
(zai;t � zai;t�1)2#1=2
: (48)
where za;t and zb;t are the k-dimensional projection of the objects a and b, and zai;t and
zbi;t each of their corresponding elements at time t.
Figure 3 plots the U.S. states synchronization map for the �rst month of the 1990�s
recession, showing that for this period of time U.S. states could be grouped into three
clusters, coinciding with the number of clusters found in Hamilton and Owyang (2010).
Speci�cally, there is a bunch of states that were more a¤ected by that recession, being
highly in sync with the national business cycle, some examples are Nevada, Ohio, Michigan,
Florida, etc., and two separate groups that experimented low synchronization with respect
to the national activity and moreover with respect to each other, in one of them are located
states like Texas, Oklahoma, Wyoming, etc., while in the other asynchronized group are
found states like Rhode Island, New Jersey, Maine, etc.7
In Figure 4 and Figure 5, the same exercise performed in the previous �gure is applied
for the 2001�s and 2007�s recessions respectively, showing that for both episodes U.S.
states can be grouped only into two clusters, which can be interpreted as the "core" and
the "periphery". The "core" that is highly in sync with the national cycle contains the
great part of the states, while the "periphery" just few of them, some examples are North
Dakota, Wyoming, Oklahoma, Montana, etc.
7The full animated representation can be found at https://sites.google.com/site/
daniloleivaleon/media
19
3.3 Clustering Coe¢ cient
The intuition behind the proposed framework for monitoring synchronization in Section 2,
relies on the fact that if �abt � 1, then it is likely that states a and b are sharing the same
business cycle dynamics, or in other words, economic shocks a¤ecting to a could also be
a¤ecting to b creating a link of dependence between them. By means of this, U.S. states,
denoted by hi for i = 1; : : : ; n, and collected in H = fhign1 , can be interpreted as nodes,
interacting in the weighted network gt, with the relationship between each pair of nodes
ha and hb at t, given by the strength of their link �abt 2 [0; 1] and collected in �t.
The previous graphs mapping states business cycle similarities for the beginning of the
last three recessions showed two di¤erent scenarios. In the �rst one, 1990�s recession, states
tend to be less synchronized between them and also with respect to the National business
cycle than the other two, 2001�s and 2007�s recessions. These changes in interconnectedness
can be monitored by using the clustering coe¢ cient for each of the n = 48 states weighted
by the degree of synchronization, CLi;t, in order to compute the clustering coe¢ cient for
the whole network at time t by averaging them, see Watts and Strogatz (1998):
CLt =1
n
nXi=1
CLi;t; (49)
where CLt would measure the statistical level of cohesiveness between the business cycle
phases of U.S. states for every period of time.
In Figure 6 is plotted the dynamic average clustering coe¢ cient, not surprisingly it
presents low values during the 1990�s recession, but high values during the 2001�s and
2007�s recessions. Moreover, that series shows that in the mid 90�s there was a change in
the cohesiveness between states since before that date it was cyclically changing from high
to low values, but after that, it remained almost stable in high values.
In order to assess if these features can be evaluated in real time for monitoring purposes,
the framework was reestimated two times with data up to one month before the beginning
of the last two recessions.8 The top and bottom charts in Figure 7 plot the index of
states cohesiveness for the 2007�s and 2001�s recessions respectively, showing a very similar
8The same analysis was not performed for the 1990�s, an the previous recessions due to the small size
of the sample.
20
performance as in the estimation with the full data set, corroborating the reliability of the
U.S. states cohesiveness index.
Another fundamental issue regarding to the business cycle synchronization literature
is the existence of a well de�ned core or atractor, which is usually represented by a leading
economy or a weighted average between a set of economies representing such core, that
could help to anticipate the movements in the aggregate business cycle. Camacho et al.
(2006) develop a speci�c procedure, based on simulations of the similarities among pairs
of economies, in order to assess if the points (economies) plotted in the business cycle
map are randomly distributed or there is any kind of atractor that keep them together.
This paper tries to explore up to which extend the interconnectedness between smaller
economies can be helpful to anticipate national recessions?
The Markov-Switching Synchronization Network (MSYN) will be used to determine the
connectedness between nodes (states) and therefore the "key" nodes (states) composing
the U.S. business cycle core or leading economies, that will help to anticipate global
recessions. In order to have a glimpse about the form that the MSYN would take during
contractionary episodes, Figure 8 plots the corresponding graph for the �rst month of
the last three recessions. Given that the MSYN is a weighted network, in order to make
possible the graphical representation, the link between a and b is plotted if �abt > 0:5,
otherwise there is no link between them. It is important to note that although the U.S.
business cycle is not included in the network analysis, just the states, all the three plots
show close relation with the ones in Figures 3, 4 and 5, respectively, where the national
cycle is included, corroborating the grouping pattern.
By looking at a disaggregated perspective, this paper has shown that U.S. states present
a high dynamic heterogeneity among their business cycle phases, due to while some states
are entering in recession, the others may or may not follow the same pattern. This study
takes advantage of this fact departing from the hypothesis that when this heterogeneity
starts to decrease, meaning that all states start to converge to the same phase, something
bad is going to happen, agreeing with the traditionally view that economies tend to show
higher converge during periods of recessions than during periods of expansion.
In order to �nd the core dynamics leading the national economy, it is required to
21
look at the prominence of each node (state) in the present synchronization network, for
this purpose the concept used to answer these kind of questions is the one of centrality.
There are several measures regarding to the centrality of a node in a network, but given
that desynchronization measures, abt , under the present framework are interpreted as
distances, providing information about the farness among U.S. states business cycles, the
most appropriate measure for this context is the one of closeness centrality.
3.4 Closeness Centrality
The farness of a given node is de�ned as the sum of its distances to all other nodes, where
the distance between two nodes i and j is given by the length of the shortest path between
them, denoted by d(i; j). In order to compute d(i; j), the Dijkstra�s (1959) algorithm is
applied since it solves the shortest path problem on a weighted graph Gt = (H;�t) for
the case in which all edges weights are nonnegative. For example, in a set H 0 = fa; b; cg
where �ab = 0:5, �ac = 0:9 and �bc = 0:2, the shortest path between a and c will be 0:7,
since �ab+ �bc < �ac, hence notice that d(a; c) does not necessarily have to be equal to �ac.
The closeness between two nodes is de�ned as the inverse of the farness. Thus, the
more central is a node, the lower is its total distance to all other nodes. Closeness can
be regarded as a measure of how fast it will take to spread information, risk, economic
shocks, etc., from node i to all other nodes sequentially.9 The total distance from node i
to all other nodes in the network g, i.e. the farness of i at time t, is given byPj 6=ijt dt(i; j),
being its reciprocal the closeness centrality of node i:
Ct(i) =1P
j 6=ijt dt(i; j): (50)
This measure of centrality has a natural analogue at the aggregate network level. Let
i� be the node that attains the highest closeness centrality across all nodes and let Ct(i�)
be this centrality at time t, then the closeness centrality of the network is given by
Ct =
kXi=1jt
[Ct(i�)� Ct(i)]: (51)
9For an overview regarding to de�nitions in network analysis, see Goyal (2007).
22
This measure is calculated for every period of time with the information available in
�t for the whole sample, i.e. since 1979:8 until 2012:3, plotting in Figure 9 the dynamics
of the closeness centrality for the MSYN involving U.S. states, showing a clear and easy to
interpret pattern. The centrality shows a markedly high tendency to increase some months
before national recessions take place, keeping high values during the whole contractionary
episode and even some months after it ends, then returning to an stable level that seems to
prevail in general during the rest of the expansion period. High centrality levels reported
after recessions have ended could be in part attributed to the so-called "third phase" in
the business cycle of some U.S. states, since the pattern associates stable growth with low
centrality, while recessions and recoveries with high centrality.
The most useful feature of the centrality index, Cc;t, is the capacity to anticipate
national recessions, since some periods before a contractionary episode begin, it starts
to increase. In order evaluate this predictive power the index was reestimated two times
with data up to one month before the beginning of the last two recessions.10 The top and
bottom charts in Figure 10 plot the centrality index for the 2007�s and 2001�s recessions
respectively, corroborating its reliability to anticipate U.S. recessions.
Additionally, by accounting for the centrality of each node (state) trough time using
Equation (50), the states with the highest centrality at each month during the sample
period can be interpreted as the atractor. The results show that there is a set of �ve
"main" states which includes, the main atractor, North Carolina with the 43.9% of times
being the state with the highest centrality, followed by Missouri with 22.7%, Wisconsin
with 21.4%, Tennessee with 11.2%, and Alabama with a 0.8%, the rest of states never
experienced the highest centrality at any period. These states are highlighted in Figure
11, showing that, with the exception of Wisconsin, they are also geographically linked,
composing a region of the U.S. business cycle sync core.
10The same analysis was not performed for the 1990�s, an the previous recessions due to the small size
of the sample.
23
3.5 Stationary Synchronization
Apart from dynamic estimates of the interdependence degree, this framework also pro-
vides the possibility of computing stationary estimates, obtained with the ergodic or
time-independent probabilities of Vt, as in Hamilton (1994), that can be interpreted as
an average sync during the period under study. Table 3 reports the matrix containing
the stationary synchronization among U.S. states, �, in which the shaded cells refer to
pairs of states with the highest sync degree (� � 0:9). The values in � range from 0.1,
which is the Delaware vs. Louisiana case, until 0.95, some examples are Kentucky vs.
Tennessee, North Carolina vs. South Carolina, and Tennessee vs. Wisconsin, which no
coincidentally contain states that belong to the core previously speci�ed. This information
can be helpful, for example, to assess which states would be more a¤ected if there is an
oil shock hitting Texas, since according to Table 3, the answer will be Colorado, followed
by Oklahoma.
The same procedure can be followed to compute stationary state vs. national synchro-
nizations, that are reported in Table 4. Not surprisingly the states in the core show high
values, that is, Missouri (0.84), Wisconsin (0.87), Alabama (0.92), Tennessee (0.93), and
the main atractor, the highest value, i.e. North Carolina (0.94). Interestingly, these states
also show high cooncordance with the National business cycle according to the results in
Owyang et al. (2005), obtained under a univariate approach. Moreover, a whole stationary
sync picture is obtained and the resulting map is reported in Figure 12, showing that for
1979:8-2012:3, U.S. states can be grouped into three clusters, a highly, discreetly and lowly
in sync with the national business cycle. This results also nests the one in Hamilton and
Owyang (2010), where it is found three clusters with individual states in a given cluster
sharing certain business cycle characteristics.
4 Conclusions
This paper proposes a novel framework for monitoring changes in the degree of aggregate
synchronization between many stochastic processes that are subject to regime changes.
Speci�cally, it computes regime inferences from a multivariate Markov-switching model
24
and simultaneously estimate a measure of the time-varying synchronization between the
unobserved state variables governing each process involved in the system.
The proposed framework is used as a tool for monitoring the time-varying synchro-
nization among U.S. states business cycles in order to assess, �rst, up to which extend
the interconnectedness between smaller economies can be helpful to anticipate national
recessions? and second, how a national recession that is coming soon is going to a¤ect
each of its smaller economies at the state level?
The results suggest that national recessions can be anticipated by an index that ac-
counts for the aggregate synchronization of states in an interdependent environment, con-
�rming its reliability with real time exercises, and that it can be evaluated how much state
"A" could be a¤ected by economic shocks hitting state "B" or the nation as a whole, since
the degree of interdependence between states can be monitored month-to-month.
Additionally, by taking into account the centrality of each state in terms of synchro-
nization, it is obtained the leading economies in the U.S. business cycle, composed by the
main atractor, North Carolina, followed by Missouri, Wisconsin, Tennessee, and Alabama.
Interestingly this set of �ve states are also geographically linked, providing a region of the
core leading the national economy that helps to anticipate contractionary episodes, and
that is located in the intersection among three of the eight BEA regions, Southeast, Great
Lakes and Plains.
25
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28
Table 1. Bivariate MS Synchronization Model for New York and Texas
Parameter Mean Std. Dev. Median
ny,0 -0.0934 0.0131 -0.0944
ny,1 0.2480 0.0126 0.2485
ny2 0.0073 0.0005 0.0072
pny,11 0.9837 0.0073 0.9847
pny,00 0.9312 0.0263 0.9346
tx,0 -0.0601 0.0102 -0.0601
tx,1 0.1691 0.0107 0.1692
tx2 0.0064 0.0004 0.0064
ptx,11 0.9839 0.0070 0.9852
ptx,00 0.9358 0.0251 0.9390
ny,tx 0.0013 0.0004 0.0013
p11 0.9808 0.0079 0.9819
p00 0.9292 0.0264 0.9323
pV,11 0.9501 0.0481 0.9647
pV,00 0.9480 0.0360 0.9565
Note: The selected example presents the case of two states with high and similar U.S. GDP share, New
York with 7.68%, and Texas with 7.95%. The results for the other cases are available upon request to
the author.
Table 2. Bivariate MS Synchronization Model for California and Vermont
Parameter Mean Std. Dev. Median
ca,0 -0.0211 0.0100 -0.0215
ca,1 0.1674 0.0103 0.1673
ca2 0.0068 0.0005 0.0068
pca,11 0.9773 0.0090 0.9785
pca,00 0.9450 0.0205 0.9472
vt,0 -0.0523 0.0136 -0.0523
vt,1 0.2012 0.0141 0.2010
vt2 0.0127 0.0010 0.0127
pvt,11 0.9764 0.0093 0.9774
pvt,00 0.9413 0.0217 0.9432
ca,vt 0.0028 0.0005 0.0027
p11 0.9772 0.0092 0.9784
p00 0.9432 0.0210 0.9458
pV,11 0.9742 0.0242 0.9817
pV,00 0.9362 0.0417 0.9461
Note: The selected example presents the case of the states with the highest and the lowest U.S. GDP
share, i.e. California with 13.34% and Vermont with 0.18%. The results for the other cases are
available upon request to the author.
29
Table 3. Stationary MS Synchronization among U.S. States Business Cycle Phases
AL AZ AR CA CO CT DE FL GA ID IL IN IA KS KY LA
AL 1 0.84 0.92 0.64 0.54 0.73 0.64 0.62 0.89 0.78 0.84 0.83 0.81 0.93 0.90 0.23
AZ 0.84 1 0.75 0.84 0.55 0.85 0.82 0.92 0.94 0.73 0.73 0.65 0.44 0.75 0.66 0.25
AR 0.92 0.75 1 0.60 0.56 0.56 0.66 0.71 0.78 0.80 0.83 0.74 0.86 0.87 0.89 0.15
CA 0.64 0.84 0.60 1 0.52 0.78 0.74 0.82 0.83 0.61 0.76 0.61 0.45 0.73 0.62 0.15
CO 0.54 0.55 0.56 0.52 1 0.57 0.27 0.45 0.59 0.69 0.58 0.36 0.66 0.61 0.51 0.53
CT 0.73 0.85 0.56 0.78 0.57 1 0.89 0.68 0.80 0.36 0.66 0.49 0.53 0.62 0.58 0.14
DE 0.64 0.82 0.66 0.74 0.27 0.89 1 0.85 0.93 0.48 0.75 0.58 0.47 0.60 0.62 0.10
FL 0.62 0.92 0.71 0.82 0.45 0.68 0.85 1 0.91 0.59 0.73 0.48 0.38 0.53 0.52 0.22
GA 0.89 0.94 0.78 0.83 0.59 0.80 0.93 0.91 1 0.61 0.81 0.56 0.60 0.73 0.70 0.14
ID 0.78 0.73 0.80 0.61 0.69 0.36 0.48 0.59 0.61 1 0.62 0.64 0.86 0.87 0.66 0.33
IL 0.84 0.73 0.83 0.76 0.58 0.66 0.75 0.73 0.81 0.62 1 0.67 0.89 0.77 0.85 0.16
IN 0.83 0.65 0.74 0.61 0.36 0.49 0.58 0.48 0.56 0.64 0.67 1 0.74 0.84 0.89 0.29
IA 0.81 0.44 0.86 0.45 0.66 0.53 0.47 0.38 0.60 0.86 0.89 0.74 1 0.88 0.92 0.24
KS 0.93 0.75 0.87 0.73 0.61 0.62 0.60 0.53 0.73 0.87 0.77 0.84 0.88 1 0.92 0.27
KY 0.90 0.66 0.89 0.62 0.51 0.58 0.62 0.52 0.70 0.66 0.85 0.89 0.92 0.92 1 0.22
LA 0.23 0.25 0.15 0.15 0.53 0.14 0.10 0.22 0.14 0.33 0.16 0.29 0.24 0.27 0.22 1
ME 0.68 0.88 0.69 0.76 0.39 0.92 0.91 0.74 0.89 0.62 0.74 0.62 0.56 0.68 0.66 0.21
MD 0.63 0.90 0.67 0.84 0.46 0.77 0.91 0.73 0.83 0.64 0.63 0.52 0.49 0.53 0.52 0.17
MA 0.53 0.70 0.55 0.64 0.43 0.92 0.80 0.59 0.83 0.19 0.65 0.39 0.50 0.49 0.48 0.13
MI 0.86 0.65 0.79 0.67 0.33 0.48 0.65 0.60 0.62 0.54 0.66 0.95 0.62 0.85 0.89 0.23
MN 0.80 0.83 0.82 0.73 0.56 0.51 0.68 0.69 0.83 0.84 0.87 0.84 0.88 0.88 0.88 0.28
MS 0.86 0.51 0.84 0.49 0.37 0.43 0.44 0.40 0.62 0.77 0.74 0.81 0.80 0.82 0.87 0.23
MO 0.91 0.89 0.92 0.69 0.55 0.70 0.89 0.85 0.93 0.72 0.92 0.76 0.79 0.91 0.93 0.13
MT 0.60 0.35 0.41 0.24 0.51 0.23 0.17 0.25 0.33 0.85 0.31 0.41 0.53 0.49 0.46 0.31
NE 0.85 0.62 0.78 0.49 0.60 0.40 0.43 0.43 0.66 0.87 0.88 0.60 0.89 0.89 0.84 0.26
NV 0.80 0.93 0.66 0.85 0.48 0.41 0.72 0.91 0.84 0.75 0.67 0.73 0.65 0.73 0.67 0.24
NH 0.71 0.92 0.69 0.73 0.59 0.94 0.87 0.81 0.87 0.50 0.64 0.36 0.58 0.62 0.46 0.16
NJ 0.66 0.93 0.67 0.85 0.50 0.94 0.91 0.90 0.87 0.55 0.61 0.52 0.52 0.64 0.54 0.18
NM 0.72 0.84 0.62 0.57 0.60 0.37 0.27 0.78 0.79 0.85 0.70 0.66 0.54 0.85 0.63 0.35
NY 0.70 0.73 0.49 0.88 0.57 0.92 0.89 0.74 0.80 0.33 0.68 0.63 0.53 0.71 0.61 0.19
NC 0.92 0.91 0.88 0.82 0.62 0.76 0.92 0.78 0.95 0.65 0.84 0.82 0.62 0.91 0.90 0.17
ND 0.32 0.23 0.22 0.16 0.43 0.26 0.14 0.20 0.14 0.42 0.26 0.33 0.36 0.35 0.33 0.30
OH 0.89 0.67 0.78 0.77 0.32 0.45 0.68 0.53 0.76 0.60 0.79 0.94 0.68 0.86 0.85 0.30
OK 0.31 0.26 0.22 0.23 0.74 0.29 0.18 0.25 0.28 0.27 0.34 0.18 0.45 0.61 0.26 0.77
OR 0.89 0.71 0.90 0.62 0.47 0.41 0.74 0.50 0.63 0.73 0.91 0.88 0.91 0.79 0.91 0.27
PA 0.87 0.78 0.64 0.79 0.42 0.73 0.80 0.73 0.81 0.59 0.88 0.84 0.77 0.83 0.89 0.26
RI 0.53 0.81 0.41 0.48 0.17 0.63 0.84 0.62 0.67 0.56 0.37 0.45 0.30 0.45 0.34 0.18
SC 0.93 0.91 0.86 0.79 0.42 0.74 0.87 0.77 0.94 0.66 0.86 0.79 0.70 0.83 0.85 0.21
SD 0.89 0.63 0.89 0.57 0.55 0.44 0.35 0.43 0.47 0.85 0.73 0.73 0.90 0.87 0.85 0.27
TN 0.94 0.88 0.90 0.75 0.49 0.66 0.67 0.67 0.90 0.69 0.84 0.93 0.84 0.91 0.95 0.22
TX 0.49 0.41 0.39 0.42 0.92 0.41 0.28 0.43 0.50 0.35 0.63 0.30 0.56 0.59 0.47 0.72
UT 0.69 0.85 0.73 0.61 0.89 0.60 0.31 0.47 0.63 0.78 0.82 0.34 0.78 0.86 0.77 0.30
VT 0.74 0.90 0.58 0.71 0.48 0.91 0.84 0.79 0.89 0.53 0.68 0.64 0.49 0.74 0.52 0.22
VA 0.81 0.93 0.76 0.79 0.53 0.86 0.95 0.88 0.95 0.63 0.83 0.57 0.56 0.68 0.62 0.15
WA 0.93 0.85 0.91 0.60 0.60 0.65 0.73 0.53 0.89 0.76 0.91 0.83 0.86 0.88 0.91 0.18
WV 0.45 0.39 0.46 0.27 0.22 0.22 0.19 0.47 0.22 0.38 0.77 0.44 0.71 0.67 0.56 0.39
WI 0.92 0.78 0.87 0.69 0.59 0.60 0.68 0.58 0.81 0.79 0.84 0.89 0.91 0.93 0.92 0.27
WY 0.27 0.26 0.16 0.17 0.30 0.15 0.14 0.24 0.16 0.25 0.16 0.25 0.27 0.27 0.25 0.82
30
Table 3 (cont.) Stationary MS Synchronization among U.S. States Business Cycle Phases
ME MD MA MI MN MS MO MT NE NV NH NJ NM NY NC ND
AL 0.68 0.63 0.53 0.86 0.80 0.86 0.91 0.60 0.85 0.80 0.71 0.66 0.72 0.70 0.92 0.32
AZ 0.88 0.90 0.70 0.65 0.83 0.51 0.89 0.35 0.62 0.93 0.92 0.93 0.84 0.73 0.91 0.23
AR 0.69 0.67 0.55 0.79 0.82 0.84 0.92 0.41 0.78 0.66 0.69 0.67 0.62 0.49 0.88 0.22
CA 0.76 0.84 0.64 0.67 0.73 0.49 0.69 0.24 0.49 0.85 0.73 0.85 0.57 0.88 0.82 0.16
CO 0.39 0.46 0.43 0.33 0.56 0.37 0.55 0.51 0.60 0.48 0.59 0.50 0.60 0.57 0.62 0.43
CT 0.92 0.77 0.92 0.48 0.51 0.43 0.70 0.23 0.40 0.41 0.94 0.94 0.37 0.92 0.76 0.26
DE 0.91 0.91 0.80 0.65 0.68 0.44 0.89 0.17 0.43 0.72 0.87 0.91 0.27 0.89 0.92 0.14
FL 0.74 0.73 0.59 0.60 0.69 0.40 0.85 0.25 0.43 0.91 0.81 0.90 0.78 0.74 0.78 0.20
GA 0.89 0.83 0.83 0.62 0.83 0.62 0.93 0.33 0.66 0.84 0.87 0.87 0.79 0.80 0.95 0.14
ID 0.62 0.64 0.19 0.54 0.84 0.77 0.72 0.85 0.87 0.75 0.50 0.55 0.85 0.33 0.65 0.42
IL 0.74 0.63 0.65 0.66 0.87 0.74 0.92 0.31 0.88 0.67 0.64 0.61 0.70 0.68 0.84 0.26
IN 0.62 0.52 0.39 0.95 0.84 0.81 0.76 0.41 0.60 0.73 0.36 0.52 0.66 0.63 0.82 0.33
IA 0.56 0.49 0.50 0.62 0.88 0.80 0.79 0.53 0.89 0.65 0.58 0.52 0.54 0.53 0.62 0.36
KS 0.68 0.53 0.49 0.85 0.88 0.82 0.91 0.49 0.89 0.73 0.62 0.64 0.85 0.71 0.91 0.35
KY 0.66 0.52 0.48 0.89 0.88 0.87 0.93 0.46 0.84 0.67 0.46 0.54 0.63 0.61 0.90 0.33
LA 0.21 0.17 0.13 0.23 0.28 0.23 0.13 0.31 0.26 0.24 0.16 0.18 0.35 0.19 0.17 0.30
ME 1 0.92 0.90 0.69 0.69 0.54 0.80 0.43 0.61 0.75 0.90 0.93 0.56 0.87 0.86 0.18
MD 0.92 1 0.76 0.59 0.75 0.37 0.82 0.31 0.54 0.66 0.79 0.92 0.37 0.82 0.75 0.20
MA 0.90 0.76 1 0.50 0.42 0.35 0.53 0.17 0.32 0.30 0.93 0.91 0.23 0.88 0.74 0.17
MI 0.69 0.59 0.50 1 0.87 0.81 0.86 0.31 0.54 0.61 0.46 0.60 0.67 0.69 0.88 0.32
MN 0.69 0.75 0.42 0.87 1 0.72 0.89 0.48 0.87 0.85 0.62 0.58 0.72 0.66 0.86 0.42
MS 0.54 0.37 0.35 0.81 0.72 1 0.84 0.57 0.75 0.78 0.41 0.33 0.71 0.52 0.86 0.27
MO 0.80 0.82 0.53 0.86 0.89 0.84 1 0.47 0.88 0.84 0.72 0.75 0.79 0.69 0.94 0.31
MT 0.43 0.31 0.17 0.31 0.48 0.57 0.47 1 0.62 0.40 0.23 0.32 0.67 0.17 0.40 0.34
NE 0.61 0.54 0.32 0.54 0.87 0.75 0.88 0.62 1 0.77 0.44 0.47 0.81 0.42 0.74 0.39
NV 0.75 0.66 0.30 0.61 0.85 0.78 0.84 0.40 0.77 1 0.67 0.72 0.91 0.46 0.78 0.23
NH 0.90 0.79 0.93 0.46 0.62 0.41 0.72 0.23 0.44 0.67 1 0.94 0.53 0.84 0.82 0.15
NJ 0.93 0.92 0.91 0.60 0.58 0.33 0.75 0.32 0.47 0.72 0.94 1 0.59 0.93 0.85 0.15
NM 0.56 0.37 0.23 0.67 0.72 0.71 0.79 0.67 0.81 0.91 0.53 0.59 1 0.37 0.81 0.29
NY 0.87 0.82 0.88 0.69 0.66 0.52 0.69 0.17 0.42 0.46 0.84 0.93 0.37 1 0.83 0.27
NC 0.86 0.75 0.74 0.88 0.86 0.86 0.94 0.40 0.74 0.78 0.82 0.85 0.81 0.83 1 0.30
ND 0.18 0.20 0.17 0.32 0.42 0.27 0.31 0.34 0.39 0.23 0.15 0.15 0.29 0.27 0.30 1
OH 0.62 0.52 0.45 0.95 0.89 0.82 0.89 0.35 0.74 0.68 0.55 0.57 0.63 0.64 0.91 0.39
OK 0.24 0.21 0.25 0.19 0.35 0.27 0.27 0.24 0.45 0.21 0.25 0.22 0.26 0.26 0.31 0.43
OR 0.74 0.65 0.44 0.85 0.89 0.85 0.77 0.42 0.79 0.60 0.48 0.59 0.56 0.63 0.79 0.34
PA 0.80 0.78 0.64 0.81 0.88 0.79 0.91 0.38 0.82 0.75 0.67 0.77 0.78 0.89 0.87 0.42
RI 0.89 0.58 0.64 0.54 0.45 0.39 0.45 0.30 0.40 0.48 0.83 0.80 0.37 0.54 0.51 0.18
SC 0.80 0.84 0.73 0.87 0.86 0.82 0.92 0.43 0.72 0.82 0.79 0.75 0.64 0.81 0.95 0.26
SD 0.64 0.54 0.38 0.68 0.88 0.77 0.75 0.76 0.90 0.60 0.45 0.41 0.71 0.54 0.75 0.38
TN 0.74 0.57 0.51 0.94 0.90 0.87 0.94 0.40 0.79 0.80 0.68 0.70 0.69 0.79 0.94 0.30
TX 0.29 0.28 0.45 0.26 0.55 0.36 0.50 0.21 0.54 0.27 0.48 0.34 0.59 0.50 0.48 0.48
UT 0.47 0.51 0.41 0.34 0.69 0.67 0.74 0.48 0.67 0.76 0.61 0.58 0.88 0.63 0.70 0.35
VT 0.91 0.79 0.87 0.75 0.70 0.52 0.81 0.29 0.58 0.69 0.91 0.94 0.55 0.83 0.89 0.24
VA 0.92 0.94 0.83 0.73 0.83 0.54 0.91 0.35 0.58 0.77 0.81 0.93 0.47 0.84 0.93 0.18
WA 0.67 0.76 0.63 0.80 0.86 0.83 0.91 0.50 0.83 0.84 0.65 0.65 0.63 0.57 0.89 0.25
WV 0.38 0.25 0.19 0.47 0.74 0.72 0.61 0.34 0.67 0.37 0.29 0.29 0.41 0.31 0.37 0.53
WI 0.73 0.46 0.52 0.90 0.92 0.87 0.93 0.43 0.91 0.77 0.60 0.55 0.82 0.74 0.93 0.44
WY 0.22 0.20 0.14 0.22 0.30 0.27 0.16 0.25 0.27 0.25 0.15 0.20 0.37 0.22 0.20 0.31
31
Table 3 (cont.) Stationary MS Synchronization among U.S. States Business Cycle Phases
OH OK OR PA RI SC SD TN TX UT VT VA WA WV WI WY
AL 0.89 0.31 0.89 0.87 0.53 0.93 0.89 0.94 0.49 0.69 0.74 0.81 0.93 0.45 0.92 0.27
AZ 0.67 0.26 0.71 0.78 0.81 0.91 0.63 0.88 0.41 0.85 0.90 0.93 0.85 0.39 0.78 0.26
AR 0.78 0.22 0.90 0.64 0.41 0.86 0.89 0.90 0.39 0.73 0.58 0.76 0.91 0.46 0.87 0.16
CA 0.77 0.23 0.62 0.79 0.48 0.79 0.57 0.75 0.42 0.61 0.71 0.79 0.60 0.27 0.69 0.17
CO 0.32 0.74 0.47 0.42 0.17 0.42 0.55 0.49 0.92 0.89 0.48 0.53 0.60 0.22 0.59 0.30
CT 0.45 0.29 0.41 0.73 0.63 0.74 0.44 0.66 0.41 0.60 0.91 0.86 0.65 0.22 0.60 0.15
DE 0.68 0.18 0.74 0.80 0.84 0.87 0.35 0.67 0.28 0.31 0.84 0.95 0.73 0.19 0.68 0.14
FL 0.53 0.25 0.50 0.73 0.62 0.77 0.43 0.67 0.43 0.47 0.79 0.88 0.53 0.47 0.58 0.24
GA 0.76 0.28 0.63 0.81 0.67 0.94 0.47 0.90 0.50 0.63 0.89 0.95 0.89 0.22 0.81 0.16
ID 0.60 0.27 0.73 0.59 0.56 0.66 0.85 0.69 0.35 0.78 0.53 0.63 0.76 0.38 0.79 0.25
IL 0.79 0.34 0.91 0.88 0.37 0.86 0.73 0.84 0.63 0.82 0.68 0.83 0.91 0.77 0.84 0.16
IN 0.94 0.18 0.88 0.84 0.45 0.79 0.73 0.93 0.30 0.34 0.64 0.57 0.83 0.44 0.89 0.25
IA 0.68 0.45 0.91 0.77 0.30 0.70 0.90 0.84 0.56 0.78 0.49 0.56 0.86 0.71 0.91 0.27
KS 0.86 0.61 0.79 0.83 0.45 0.83 0.87 0.91 0.59 0.86 0.74 0.68 0.88 0.67 0.93 0.27
KY 0.85 0.26 0.91 0.89 0.34 0.85 0.85 0.95 0.47 0.77 0.52 0.62 0.91 0.56 0.92 0.25
LA 0.30 0.77 0.27 0.26 0.18 0.21 0.27 0.22 0.72 0.30 0.22 0.15 0.18 0.39 0.27 0.82
ME 0.62 0.24 0.74 0.80 0.89 0.80 0.64 0.74 0.29 0.47 0.91 0.92 0.67 0.38 0.73 0.22
MD 0.52 0.21 0.65 0.78 0.58 0.84 0.54 0.57 0.28 0.51 0.79 0.94 0.76 0.25 0.46 0.20
MA 0.45 0.25 0.44 0.64 0.64 0.73 0.38 0.51 0.45 0.41 0.87 0.83 0.63 0.19 0.52 0.14
MI 0.95 0.19 0.85 0.81 0.54 0.87 0.68 0.94 0.26 0.34 0.75 0.73 0.80 0.47 0.90 0.22
MN 0.89 0.35 0.89 0.88 0.45 0.86 0.88 0.90 0.55 0.69 0.70 0.83 0.86 0.74 0.92 0.30
MS 0.82 0.27 0.85 0.79 0.39 0.82 0.77 0.87 0.36 0.67 0.52 0.54 0.83 0.72 0.87 0.27
MO 0.89 0.27 0.77 0.91 0.45 0.92 0.75 0.94 0.50 0.74 0.81 0.91 0.91 0.61 0.93 0.16
MT 0.35 0.24 0.42 0.38 0.30 0.43 0.76 0.40 0.21 0.48 0.29 0.35 0.50 0.34 0.43 0.25
NE 0.74 0.45 0.79 0.82 0.40 0.72 0.90 0.79 0.54 0.67 0.58 0.58 0.83 0.67 0.91 0.27
NV 0.68 0.21 0.60 0.75 0.48 0.82 0.60 0.80 0.27 0.76 0.69 0.77 0.84 0.37 0.77 0.25
NH 0.55 0.25 0.48 0.67 0.83 0.79 0.45 0.68 0.48 0.61 0.91 0.81 0.65 0.29 0.60 0.15
NJ 0.57 0.22 0.59 0.77 0.80 0.75 0.41 0.70 0.34 0.58 0.94 0.93 0.65 0.29 0.55 0.20
NM 0.63 0.26 0.56 0.78 0.37 0.64 0.71 0.69 0.59 0.88 0.55 0.47 0.63 0.41 0.82 0.37
NY 0.64 0.26 0.63 0.89 0.54 0.81 0.54 0.79 0.50 0.63 0.83 0.84 0.57 0.31 0.74 0.22
NC 0.91 0.31 0.79 0.87 0.51 0.95 0.75 0.94 0.48 0.70 0.89 0.93 0.89 0.37 0.93 0.20
ND 0.39 0.43 0.34 0.42 0.18 0.26 0.38 0.30 0.48 0.35 0.24 0.18 0.25 0.53 0.44 0.31
OH 1 0.17 0.86 0.84 0.45 0.84 0.76 0.93 0.29 0.36 0.75 0.74 0.82 0.62 0.92 0.26
OK 0.17 1 0.22 0.33 0.16 0.24 0.35 0.22 0.83 0.42 0.27 0.22 0.31 0.48 0.22 0.81
OR 0.86 0.22 1 0.76 0.43 0.86 0.89 0.87 0.30 0.59 0.66 0.73 0.89 0.44 0.89 0.23
PA 0.84 0.33 0.76 1 0.63 0.86 0.70 0.92 0.52 0.62 0.84 0.86 0.82 0.71 0.92 0.30
RI 0.45 0.16 0.43 0.63 1 0.65 0.49 0.48 0.17 0.19 0.84 0.69 0.41 0.32 0.51 0.18
SC 0.84 0.24 0.86 0.86 0.65 1 0.76 0.94 0.41 0.54 0.76 0.94 0.90 0.41 0.90 0.24
SD 0.76 0.35 0.89 0.70 0.49 0.76 1 0.82 0.42 0.64 0.46 0.51 0.84 0.55 0.89 0.26
TN 0.93 0.22 0.87 0.92 0.48 0.94 0.82 1 0.40 0.73 0.74 0.81 0.91 0.52 0.95 0.27
TX 0.29 0.83 0.30 0.52 0.17 0.41 0.42 0.40 1 0.76 0.35 0.33 0.36 0.34 0.59 0.61
UT 0.36 0.42 0.59 0.62 0.19 0.54 0.64 0.73 0.76 1 0.60 0.51 0.85 0.51 0.87 0.33
VT 0.75 0.27 0.66 0.84 0.84 0.76 0.46 0.74 0.35 0.60 1 0.88 0.59 0.35 0.76 0.18
VA 0.74 0.22 0.73 0.86 0.69 0.94 0.51 0.81 0.33 0.51 0.88 1 0.80 0.21 0.77 0.16
WA 0.82 0.31 0.89 0.82 0.41 0.90 0.84 0.91 0.36 0.85 0.59 0.80 1 0.36 0.92 0.20
WV 0.62 0.48 0.44 0.71 0.32 0.41 0.55 0.52 0.34 0.51 0.35 0.21 0.36 1 0.77 0.47
WI 0.92 0.22 0.89 0.92 0.51 0.90 0.89 0.95 0.59 0.87 0.76 0.77 0.92 0.77 1 0.31
WY 0.26 0.81 0.23 0.30 0.18 0.24 0.26 0.27 0.61 0.33 0.18 0.16 0.20 0.47 0.31 1
32
Table 4. Stationary MS Synchronization between State and U.S. business cycle phases
State Sync State Sync State Sync State Sync
Alabama 0.92 Iowa 0.64 Nebraska 0.61 Rhode Island 0.58
Arizona 0.79 Kansas 0.85 Nevada 0.75 S. Carolina 0.92
Arkansas 0.71 Kentucky 0.84 N. Hampshire 0.60 S. Dakota 0.71
California 0.87 Louisiana 0.24 New Jersey 0.78 Tennessee 0.93
Colorado 0.41 Maine 0.80 New Mexico 0.75 Texas 0.38
Connecticut 0.66 Maryland 0.48 New York 0.77 Utah 0.72
Delaware 0.80 Massachusetts 0.44 N. Carolina 0.94 Vermont 0.85
Florida 0.68 Michigan 0.90 N. Dakota 0.32 Virginia 0.90
Georgia 0.80 Minnesota 0.90 Ohio 0.92 Washington 0.75
Idaho 0.62 Mississippi 0.85 Oklahoma 0.22 W. Virginia 0.55
Illinois 0.86 Missouri 0.84 Oregon 0.74 Wisconsin 0.87
Indiana 0.92 Montana 0.39 Pennsylvania 0.88 Wyoming 0.28
Note: The table reports the stationary degree of synchronization for 1979:8 to 2012:3. Those estimates
correspond to the ergodic probability that the business cycle phases of the corresponding state and the one of
the U.S. economic activity are totally dependent, i.e. Pr(Vt=1).
33
Figure 1. MS Synchronization between Selected States
0
1
1979 1984 1989 1994 1999 2004 2009
0
1
1979 1984 1989 1994 1999 2004 2009
0
1
1979 1984 1989 1994 1999 2004 2009
Probability of Recession in New York
Probability of Recession in Texas
New York vs. Texas Synchronization
0
1
1979 1984 1989 1994 1999 2004 2009
0
1
1979 1984 1989 1994 1999 2004 2009
0
1
1979 1984 1989 1994 1999 2004 2009
Probability of Recession in California
Probability of Recession in Vermont
California vs. Vermont Synchronization
Note. The Chart A of the figure plots the probabilities of recession for New York and Texas and its time-
varying synchronization. The Chart B of the figure plots the probabilities of recession for California and
Vermont and its time-varying synchronization. Shaded areas correspond to NBER recessions. The full set of
pairwise cases can be available upon request to the author.
34
Chart A Chart B
Figure 2. MS Synchronization between States and U.S. Economic Activity
Alabama
Note. Each chart plots the time-varying degree of synchronization between the U.S. business cycle
phases and the ones of each State. Shaded areas correspond to NBER-referenced recessions.
0
1
1979 1984 1989 1994 1999 2004 2009
Arizona
0
1
1979 1984 1989 1994 1999 2004 2009
Arkansas
0
1
1979 1984 1989 1994 1999 2004 2009
California
0
1
1979 1984 1989 1994 1999 2004 2009
Colorado
0
1
1979 1984 1989 1994 1999 2004 2009
Connecticut
0
1
1979 1984 1989 1994 1999 2004 2009
Delaware
0
1
1979 1984 1989 1994 1999 2004 2009
Florida
0
1
1979 1984 1989 1994 1999 2004 2009
Georgia
0
1
1979 1984 1989 1994 1999 2004 2009
Idaho
0
1
1979 1984 1989 1994 1999 2004 2009
Illinois
0
1
1979 1984 1989 1994 1999 2004 2009
Indiana
0
1
1979 1984 1989 1994 1999 2004 2009
35
Iowa
0
1
1979 1984 1989 1994 1999 2004 2009
Kansas
0
1
1979 1984 1989 1994 1999 2004 2009
Kentucky
0
1
1979 1984 1989 1994 1999 2004 2009
Louisiana
0
1
1979 1984 1989 1994 1999 2004 2009
Maine
0
1
1979 1984 1989 1994 1999 2004 2009
Maryland
0
1
1979 1984 1989 1994 1999 2004 2009
Massachusetts
0
1
1979 1984 1989 1994 1999 2004 2009
Michigan
0
1
1979 1984 1989 1994 1999 2004 2009
Minnesota
0
1
1979 1984 1989 1994 1999 2004 2009
Mississippi
0
1
1979 1984 1989 1994 1999 2004 2009
Missouri
0
1
1979 1984 1989 1994 1999 2004 2009
Montana
0
1
1979 1984 1989 1994 1999 2004 2009
36
Figure 2 (continued). MS Synchronization between States and U.S. Economic Activity
Note. Each chart plots the time-varying degree of synchronization between the U.S. business cycle
phases and the ones of each State. Shaded areas correspond to NBER-referenced recessions.
Nebraska
0
1
1979 1984 1989 1994 1999 2004 2009
Nevada
0
1
1979 1984 1989 1994 1999 2004 2009
New Hampshire
0
1
1979 1984 1989 1994 1999 2004 2009
New Jersey
0
1
1979 1984 1989 1994 1999 2004 2009
New Mexico
0
1
1979 1984 1989 1994 1999 2004 2009
New York
0
1
1979 1984 1989 1994 1999 2004 2009
North Carolina
0
1
1979 1984 1989 1994 1999 2004 2009
North Dakota
0
1
1979 1984 1989 1994 1999 2004 2009
Ohio
0
1
1979 1984 1989 1994 1999 2004 2009
Oklahoma
0
1
1979 1984 1989 1994 1999 2004 2009
Oregon
0
1
1979 1984 1989 1994 1999 2004 2009
Pennsylvania
0
1
1979 1984 1989 1994 1999 2004 2009
37
Figure 2 (continued). MS Synchronization between States and U.S. Economic Activity
Note. Each chart plots the time-varying degree of synchronization between the U.S. business cycle
phases and the ones of each State. Shaded areas correspond to NBER-referenced recessions.
Rhode Island
0
1
1979 1984 1989 1994 1999 2004 2009
South Carolina
0
1
1979 1984 1989 1994 1999 2004 2009
South Dakota
0
1
1979 1984 1989 1994 1999 2004 2009
Tennessee
0
1
1979 1984 1989 1994 1999 2004 2009
Texas
0
1
1979 1984 1989 1994 1999 2004 2009
Utah
0
1
1979 1984 1989 1994 1999 2004 2009
Vermont
0
1
1979 1984 1989 1994 1999 2004 2009
Virginia
0
1
1979 1984 1989 1994 1999 2004 2009
Washington
0
1
1979 1984 1989 1994 1999 2004 2009
West Virginia
0
1
1979 1984 1989 1994 1999 2004 2009
Wisconsin
0
1
1979 1984 1989 1994 1999 2004 2009
Wyoming
0
1
1979 1984 1989 1994 1999 2004 2009
38
Figure 2 (continued). MS Synchronization between States and U.S. Economic Activity
Note. Each chart plots the time-varying degree of synchronization between the U.S. business cycle
phases and the ones of each State. Shaded areas correspond to NBER-referenced recessions.
Fig
ure
3. U
.S.
Sta
tes
Syn
chro
niz
ati
on
Map
for
the
1990’s
rec
essi
on
Note
. T
he
figu
re p
lots
th
e m
ult
idim
ensi
onal
sca
ling m
ap b
ased
on
the
Eucl
idea
n d
ista
nce
of
the
U.S
. S
tate
s b
usi
nes
s
cycl
e ch
arac
teri
stic
s fo
r Ju
ly 1
99
0,
whic
h i
s th
e beg
innin
g o
f a
rece
ssio
n.
The
dis
tance
s ar
e n
orm
aliz
ed w
ith r
esp
ect
to t
he
U.S
. N
atio
nal
Eco
nom
ic A
ctiv
ity,
the
red
poin
t in
the
cente
r. T
he
size
of
the
poin
ts m
akes
ref
eren
ce t
o t
he
GD
P
shar
e o
f th
e co
rres
po
nd
ing s
tate
. T
he
full
anim
ated
ver
sion
is
avai
lable
upon
req
ues
t to
the
auth
or.
39
Fig
ure
4. U
.S.
Sta
tes
Syn
chro
niz
ati
on
Map
for
the
2001’s
rec
essi
on
40
Note
. T
he
figu
re p
lots
th
e m
ult
idim
ensi
onal
sca
ling m
ap b
ased
on
the
Eucl
idea
n d
ista
nce
of
the
U.S
. S
tate
s b
usi
nes
s
cycl
e ch
arac
teri
stic
s fo
r M
arch
2001
, w
hic
h i
s th
e beg
innin
g o
f a
rece
ssio
n.
The
dis
tance
s ar
e n
orm
aliz
ed w
ith r
esp
ect
to t
he
U.S
. N
atio
nal
Eco
nom
ic A
ctiv
ity,
the
red
poin
t in
the
cente
r. T
he
size
of
the
poin
ts m
akes
ref
eren
ce t
o t
he
GD
P
shar
e o
f th
e co
rres
po
nd
ing s
tate
. T
he
full
anim
ated
ver
sion
is
avai
lable
upon
req
ues
t to
the
auth
or.
Fig
ure
5. U
.S. S
tate
s S
yn
chro
niz
ati
on
Map
for
the
2007’s
rec
essi
on
41
Note
. T
he
figu
re p
lots
th
e m
ult
idim
ensi
onal
sca
ling m
ap b
ased
on
the
Eucl
idea
n d
ista
nce
of
the
U.S
. S
tate
s b
usi
nes
s
cycl
e ch
arac
teri
stic
s fo
r D
ecem
ber
2007
, w
hic
h i
s th
e beg
innin
g o
f th
e so
cal
led
“great
recession”.
The
dis
tan
ces
are
norm
aliz
ed w
ith r
espec
t to
the
U.S
. N
atio
nal
Eco
no
mic
Act
ivit
y,
the
red p
oin
t in
th
e ce
nte
r. T
he
size
of
the
poin
ts
mak
es r
efer
ence
to t
he
GD
P s
har
e of
the
corr
espondin
g s
tate
. T
he
full
anim
ated
ver
sion
is
avai
lable
up
on r
equ
est
to
the
auth
or.
Figure 6. Cluster Coefficient
42
0,16
0,26
0,36
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009
0,14
0,24
0,34
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009
0,14
0,24
0,34
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009
Figure 7. Real Time Estimates of Closeness Centrality
Note. The figure plots the cluster coefficient of the synchronization network composed by the U.S.
State for each period of time. Shaded areas correspond to recessions as documented by the NBER.
Note. The top chart plots the dynamic cluster coefficient with the data until 12/2007. The bottom
chart plots the dynamic cluster coefficient with the data until 03/2001. Shaded areas correspond to
recessions as documented by the NBER.
Figure 8. Synchronization Network for the Beginning of Recessions
Note. The figure plots the interconnectedness in terms of synchronization between the business cycle
phases of U.S. States. Each node represent a State, and the green each green line represent the link
between two states, which take place only if Pr(Vt=1)>0,5. The full animated version is available
upon request to the author.
43
Figure 10. Real Time Estimates of Closeness Centrality
0,8
1,3
1,8
2,3
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009
Note. The top chart plots the dynamic closeness centrality with the data until 12/2007. The bottom
chart plots the dynamic closeness centrality with the data until 03/2001. Shaded areas correspond to
recessions as documented by the NBER.
44
1
1,5
2
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009
1
2
3
4
1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009
Figure 9. Closeness Centrality
Note. The figure plots the closeness centrality of the synchronization network composed by the U.S.
state for each period of time. Shaded areas correspond to recessions as documented by the NBER.
Fig
ure
11
. U
.S.
Sta
tes
wit
h H
igh
est
Cen
trali
ty d
uri
ng 1
979:8
– 2
01
2:3
Note
. T
he
stat
es i
nsi
de
the
mar
ked
are
a co
rres
pond t
o A
labam
a, M
isso
uri
, N
ort
h C
aroli
na
and T
ennes
see,
wh
ich
are
the
stat
es
that
hav
e sh
ow
n
the
hig
hes
t cl
ose
nes
s ce
ntr
alit
y
in
the
Mar
kov
-sw
itch
ing
syn
chro
niz
atio
n
net
wo
rk
alte
rnat
ing a
mo
ng p
erio
ds
du
rin
g 1
979
:8 u
nti
l 2012
:3.
45
Fig
ure
12.
U.S
. S
tate
s S
tati
on
ary
Syn
chro
niz
ati
on
Map
Note
. T
he
figure
plo
ts t
he
mu
ltid
imen
sional
sca
ling m
ap b
ased
on
the
stat
ionar
y E
ucl
idea
n d
ista
nce
of
the
U.S
. S
tate
s
busi
nes
s cy
cle
char
acte
rist
ics
for
the
sam
ple
under
stu
dy,
Au
gust
1979
-Mar
ch 2
010
., T
he
dis
tance
s ar
e n
orm
aliz
ed
wit
h r
esp
ect
to t
he
U.S
. N
atio
nal
Eco
nom
ic A
ctiv
ity,
the
red p
oin
t in
the
cente
r. T
he
size
of
the
poin
ts m
ake
refe
ren
ce
to t
he
GD
P s
har
e o
f th
e co
rres
po
nd
ing s
tate
. T
he
full
anim
ated
ver
sion
is
avai
lable
upon
req
ues
t to
th
e au
tho
r.
46